Average Error: 0.0 → 0.0
Time: 9.5s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1853765 = x_re;
        double r1853766 = y_im;
        double r1853767 = r1853765 * r1853766;
        double r1853768 = x_im;
        double r1853769 = y_re;
        double r1853770 = r1853768 * r1853769;
        double r1853771 = r1853767 + r1853770;
        return r1853771;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1853772 = x_re;
        double r1853773 = y_im;
        double r1853774 = x_im;
        double r1853775 = y_re;
        double r1853776 = r1853774 * r1853775;
        double r1853777 = fma(r1853772, r1853773, r1853776);
        return r1853777;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 0.0

    \[x.re \cdot y.im + x.im \cdot y.re\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))